muthu commit

1 files changed, 15 insertions, 20 deletions

diff --git a/problem.tex b/problem.tex
index 4835be4..527c34a 100644
--- a/problem.tex
+++ b/problem.tex
@@ -68,7 +68,7 @@ no positive transfers ($p_i(c)\geq 0$).
 
 In addition to the above, mechanism design in budget feasible reverse auctions seeks mechanisms that have the following properties \cite{singer-mechanisms,chen}:
 \begin{itemize}
-    \item \emph{Truthfulness.} An agent has no incentive to misreport her true cost. Formally, let $c_{-i}$
+    \item \emph{Truthfulness.} An agent has no incentive to missreport her true cost. Formally, let $c_{-i}$
  be a vector of costs of all agents except $i$. %    $c_{-i}$ being the same). Let $[p_i']_{i\in \mathcal{N}} = p(c_i',
 %    c_{-i})$, then the 
 A mechanism is truthful iff for every $i \in \mathcal{N}$ and every two cost vectors $(c_i,c_{-i})$ and $(c_i',c_{-i})$
@@ -130,25 +130,7 @@ biometric information (\emph{e.g.}, her red cell blood count, a genetic marker,
 etc.). The cost $c_i$ is the amount the subject deems sufficient to
 incentivize her participation in the study. Note that, in this setup, the feature vectors $x_i$ are public information that the experimenter can consult prior the experiment design. Moreover, though a subject may lie about her true cost $c_i$, she cannot lie about $x_i$ (\emph{i.e.}, all features are verifiable upon collection) or $y_i$ (\emph{i.e.}, she cannot falsify her measurement).
 
-%\subsection{D-Optimality Criterion}
-<<<<<<< HEAD
-Ideally, motivated by the $D$-optimality criterion, we would like to design a mechanism that maximizes or approximates \eqref{dcrit} . Since \eqref{dcrit} may take arbitrarily small negative values, to define a meaningful approximation one would consider  the (equivalent) maximization of $V(S) = \det\T{X_S}X_S$. %, for some strictly increasing, on-to function $f:\reals_+\to\reals_+$. 
-However, the following lower bound implies that such an optimization goal cannot be attained under the constraints of  truthfulness, budget feasibility, and individual rationality.
-\begin{lemma}
-For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individionally rational mechanism for a budget feasible reverse auction with $V(S) =  \det{\T{X_S}X_S}$.
-=======
 Ideally, motivated by the $D$-optimality criterion, we would like to design a mechanism that maximizes \eqref{dcrit} within a good approximation ratio.  In what follows, we consider a slightly more general objective as follows:
-\junk{
-As \eqref{dcrit} may take arbitrarily small negative values, to define a meaningful approximation one would consider  the (equivalent) maximization of $V(S) = \det\T{X_S}X_S$. %, for some strictly increasing, on-to function $f:\reals_+\to\reals_+$. 
-However, the following lower bound implies that such an optimization goal cannot be attained under the constraints of  truthfulness, budget feasibility, and individual rationality.
-\begin{lemma}
-For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individually rational mechanism for a budget feasible reverse auction with value function $V(S) = \det{\T{X_S}X_S}$.
->>>>>>> c29302b25adf190f98019eb8ce5f79b10b66d54d
-\end{lemma}
-\begin{proof}
-\input{proof_of_lower_bound1}
-\end{proof}
-This negative result motivates us to study a problem with a modified objective:}
 \begin{center}
 \textsc{ExperimentalDesignProblem} (EDP)
 \begin{subequations}
@@ -158,17 +140,30 @@ This negative result motivates us to study a problem with a modified objective:}
 \end{align}\label{edp}
 \end{subequations}
 \end{center} where $I_d\in \reals^{d\times d}$ is the identity matrix.
-One possible interpretation of \eqref{modified} is that, prior to the auction, the experimenter has free access to $d$ experiments whose features form an orthonormal basis in $\reals^d$. However, \eqref{modified} can also be motivated in the context of \emph{Bayesian experimental design} \cite{chaloner1995bayesian}. In short, the objective \eqref{modified} arises naturally when the experimenter retrieves the model $\beta$ through \emph{ridge regression}, rather than the linear regression \eqref{leastsquares} over the observed data; we explore this connection in Section~\ref{sec:bed}. 
+\junk{One possible interpretation of \eqref{modified} is that, prior to the auction, the experimenter has free access to $d$ experiments whose features form an orthonormal basis in $\reals^d$. However, \eqref{modified} can also be motivated in the context of \emph{Bayesian experimental design} \cite{chaloner1995bayesian}. In short, the objective \eqref{modified} arises naturally when the experimenter retrieves the model $\beta$ through \emph{ridge regression}, rather than the linear regression \eqref{leastsquares} over the observed data; we explore this connection in Section~\ref{sec:bed}. 
 }
 
 We present our results with this version of the objective function because it is simple and captures the versions
 we will need later (including  an arbitrary matrix in place of $I_d$ or a zero matrix which will correspond to ~\eqref{dcrit} ).
+
 Note that maximizing \eqref{modified} is equivalent to maximizing \eqref{dcrit} in the full-information case. In particular, $\det(\T{X_S}X_S)> \det(\T{X_{S'}}X_{S'})$ iff $\det(I_d+\T{X_S}X_S)>\det(I_d+\T{X_{S'}}X_{S'})$. In addition, \eqref{edp}---and the equivalent problem with objective \eqref{dcrit}---are NP-hard; to see this, note that \textsc{Knapsack} reduces to EDP with dimension $d=1$ by mapping the weight of each item $w_i$ to an experiment with $x_i=w_i$. Nevertheless, \eqref{modified} is submodular, monotone and satisfies $V(\emptyset) = 0$.
 %, allowing us to use the extensive machinery for the optimization of submodular functions, as well as recent results in the 
 % context of budget feasible auctions \cite{chen,singer-mechanisms}. 
 
 %\stratis{A potential problem is that all of the above properties hold also for, \emph{e.g.}, $V(S)=\log(1+\det(\T{X_S}X_S))$\ldots}
 
+\begin{comment}\junk{
+As \eqref{dcrit} may take arbitrarily small negative values, to define a meaningful approximation one would consider  the (equivalent) maximization of $V(S) = \det\T{X_S}X_S$. %, for some strictly increasing, on-to function $f:\reals_+\to\reals_+$. 
+However, the following lower bound implies that such an optimization goal cannot be attained under the constraints of  truthfulness, budget feasibility, and individual rationality.
+\begin{lemma}
+For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individually rational mechanism for a budget feasible reverse auction with value function $V(S) = \det{\T{X_S}X_S}$.
+For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individually rational mechanism for a budget feasible reverse auction with $V(S) =  \det{\T{X_S}X_S}$.
+\end{lemma}
+\begin{proof}
+\input{proof_of_lower_bound1}
+\end{proof}
+This negative result motivates us to study a problem with a modified objective:}\end{comment}
+
 
 \begin{comment}
 \subsection{Experimental Design}